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SHOGUN
4.1.0
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This class implements the quadratic time Maximum Mean Statistic as described in [1]. The MMD is the distance of two probability distributions \(p\) and \(q\) in a RKHS which we denote by
\[ \hat{\eta_k}=\text{MMD}[\mathcal{F},p,q]^2=\textbf{E}_{x,x'} \left[ k(x,x')\right]-2\textbf{E}_{x,y}\left[ k(x,y)\right] +\textbf{E}_{y,y'}\left[ k(y,y')\right]=||\mu_p - \mu_q||^2_\mathcal{F} \]
.
Given two sets of samples \(\{x_i\}_{i=1}^{n_x}\sim p\) and \(\{y_i\}_{i=1}^{n_y}\sim q\), \(n_x+n_y=n\), the unbiased estimate of the above statistic is computed as
\[ \hat{\eta}_{k,U}=\frac{1}{n_x(n_x-1)}\sum_{i=1}^{n_x}\sum_{j\neq i} k(x_i,x_j)+\frac{1}{n_y(n_y-1)}\sum_{i=1}^{n_y}\sum_{j\neq i}k(y_i,y_j) -\frac{2}{n_xn_y}\sum_{i=1}^{n_x}\sum_{j=1}^{n_y}k(x_i,y_j) \]
A biased version is
\[ \hat{\eta}_{k,V}=\frac{1}{n_x^2}\sum_{i=1}^{n_x}\sum_{j=1}^{n_x} k(x_i,x_j)+\frac{1}{n_y^2}\sum_{i=1}^{n_y}\sum_{j=1}^{n_y}k(y_i,y_j) -\frac{2}{n_xn_y}\sum_{i=1}^{n_x}\sum_{j=1}^{n_y}k(x_i,y_j) \]
When \(n_x=n_y=\frac{n}{2}\), an incomplete version can also be computed as the following
\[ \hat{\eta}_{k,U^-}=\frac{1}{\frac{n}{2}(\frac{n}{2}-1)}\sum_{i\neq j} h(z_i,z_j) \]
where for each pair \(z=(x,y)\), \(h(z,z')=k(x,x')+k(y,y')-k(x,y')- k(x',y)\).
The type (biased/unbiased/incomplete) can be selected via set_statistic_type(). Note that there are presently two setups for computing statistic. While using BIASED, UNBIASED or INCOMPLETE, the estimate returned by compute_statistic() is \(\frac{n_xn_y}{n_x+n_y}\hat{\eta}_k\). If DEPRECATED ones are used, then this returns \((n_x+n_y)\hat{\eta}_k\) in general and \((\frac{n}{2}) \hat{\eta}_k\) when \(n_x=n_y=\frac{n}{2}\). This holds for the null distribution samples as well.
Estimating variance of the asymptotic distribution of the statistic under null and alternative hypothesis can be done using compute_variance() method. This is internally done alongwise computing statistics to avoid recomputing the kernel.
Variance under null is computed as \(\sigma_{k,0}^2=2\hat{\kappa}_2=2(\kappa_2-2\kappa_1+\kappa_0)\) where \(\kappa_0=\left(\mathbb{E}_{X,X'}k(X,X')\right )^2\), \(\kappa_1=\mathbb{E}_X\left[(\mathbb{E}_{X'}k(X,X'))^2\right]\), and \(\kappa_2=\mathbb{E}_{X,X'}k^2(X,X')\) and variance under alternative is computed as
\[ \sigma_{k,A}^2=4\rho_y\left\{\mathbb{E}_X\left[\left(\mathbb{E}_{X'} k(X,X')-\mathbb{E}_Yk(X,Y)\right)^2 \right ] -\left(\mathbb{E}_{X,X'} k(X,X')-\mathbb{E}_{X,Y}k(X,Y) \right)^2\right \}+4\rho_x\left\{ \mathbb{E}_Y\left[\left(\mathbb{E}_{Y'}k(Y,Y')-\mathbb{E}_Xk(X,Y) \right)^2\right ] -\left(\mathbb{E}_{Y,Y'}k(Y,Y')-\mathbb{E}_{X,Y} k(X,Y) \right)^2\right \} \]
where \(\rho_x=\frac{n_x}{n}\) and \(\rho_y=\frac{n_y}{n}\).
Note that statistic and variance estimation can be done for multiple kernels at once as well.
Along with the statistic comes a method to compute a p-value based on different methods. Permutation test is also possible. If unsure which one to use, sampling with 250 permutation iterations always is correct (but slow).
To choose, use set_null_approximation_method() and choose from.
MMD2_SPECTRUM_DEPRECATED: For a fast, consistent test based on the spectrum of the kernel matrix, as described in [2]. Only supported if Eigen3 is installed.
MMD2_SPECTRUM: Similar to the deprecated version except it estimates the statistic under null as \(\frac{n_xn_y}{n_x+n_y}\hat{\eta}_{k,U}\rightarrow \sum_r\lambda_r(Z_r^2-1)\) instead (see method description for more details).
MMD2_GAMMA: for a very fast, but not consistent test based on moment matching of a Gamma distribution, as described in [2].
PERMUTATION: For permuting available samples to sample null-distribution
If you do not know about your data, but want to use the MMD from a kernel matrix, just use the custom kernel constructor. Everything else will work as usual.
For kernel selection see CMMDKernelSelection.
NOTE: \(n_x\) and \(n_y\) are represented by \(m\) and \(n\), respectively in the implementation.
[1]: Gretton, A., Borgwardt, K. M., Rasch, M. J., Schoelkopf, B., & Smola, A. (2012). A Kernel Two-Sample Test. Journal of Machine Learning Research, 13, 671-721.
[2]: Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011). A fast, consistent kernel two-sample test.
在文件 QuadraticTimeMMD.h 第 158 行定义.
Public 属性 | |
| SGIO * | io |
| Parallel * | parallel |
| Version * | version |
| Parameter * | m_parameters |
| Parameter * | m_model_selection_parameters |
| Parameter * | m_gradient_parameters |
| uint32_t | m_hash |
Protected 成员函数 | |
| SGVector< float64_t > | compute_unbiased_statistic_variance (int m, int n) |
| SGVector< float64_t > | compute_biased_statistic_variance (int m, int n) |
| SGVector< float64_t > | compute_incomplete_statistic_variance (int n) |
| float64_t | compute_unbiased_statistic (int m, int n) |
| float64_t | compute_biased_statistic (int m, int n) |
| float64_t | compute_incomplete_statistic (int n) |
| virtual void | load_serializable_pre () throw (ShogunException) |
| virtual void | load_serializable_post () throw (ShogunException) |
| virtual void | save_serializable_pre () throw (ShogunException) |
| virtual void | save_serializable_post () throw (ShogunException) |
default constructor
在文件 QuadraticTimeMMD.cpp 第 48 行定义.
| CQuadraticTimeMMD | ( | CKernel * | kernel, |
| CFeatures * | p_and_q, | ||
| index_t | m | ||
| ) |
Constructor
| p_and_q | feature data. Is assumed to contain samples from both p and q. First m samples from p, then from index m all samples from q |
| kernel | kernel to use |
| p_and_q | samples from p and q, appended |
| m | index of first sample of q |
在文件 QuadraticTimeMMD.cpp 第 53 行定义.
| CQuadraticTimeMMD | ( | CKernel * | kernel, |
| CFeatures * | p, | ||
| CFeatures * | q | ||
| ) |
Constructor. This is a convienience constructor which copies both features to one element and then calls the other constructor. Needs twice the memory for a short time
| kernel | kernel for MMD |
| p | samples from distribution p, will be copied and NOT SG_REF'ed |
| q | samples from distribution q, will be copied and NOT SG_REF'ed |
在文件 QuadraticTimeMMD.cpp 第 60 行定义.
| CQuadraticTimeMMD | ( | CCustomKernel * | custom_kernel, |
| index_t | m | ||
| ) |
Constructor. This is a convienience constructor which allows to only specify a custom kernel. In this case, the features are completely ignored and all computations will be done on the custom kernel
| custom_kernel | custom kernel for MMD, which is a kernel between the appended features p and q |
| m | index of first sample of q |
在文件 QuadraticTimeMMD.cpp 第 66 行定义.
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virtual |
destructor
在文件 QuadraticTimeMMD.cpp 第 72 行定义.
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Builds a dictionary of all parameters in SGObject as well of those of SGObjects that are parameters of this object. Dictionary maps parameters to the objects that own them.
| dict | dictionary of parameters to be built. |
在文件 SGObject.cpp 第 597 行定义.
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Creates a clone of the current object. This is done via recursively traversing all parameters, which corresponds to a deep copy. Calling equals on the cloned object always returns true although none of the memory of both objects overlaps.
在文件 SGObject.cpp 第 714 行定义.
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Wrapper method for computing biased estimate of MMD^2
| m | number of samples from p |
| n | number of samples from q |
在文件 QuadraticTimeMMD.cpp 第 536 行定义.
Helper method to compute biased estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis
| m | number of samples from p |
| n | number of samples from q |
在文件 QuadraticTimeMMD.cpp 第 239 行定义.
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Wrapper method for computing incomplete estimate of MMD^2
| n | number of samples from p and q |
在文件 QuadraticTimeMMD.cpp 第 541 行定义.
Helper method to compute incomplete estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis
| n | number of samples from p and q |
在文件 QuadraticTimeMMD.cpp 第 385 行定义.
computes a p-value based on current method for approximating the null-distribution. The p-value is the 1-p quantile of the null- distribution where the given statistic lies in.
Not all methods for computing the p-value are compatible with all methods of computing the statistic (biased/unbiased/incomplete).
| statistic | statistic value to compute the p-value for |
重载 CTwoSampleTest .
在文件 QuadraticTimeMMD.cpp 第 749 行定义.
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virtual |
Computes the squared quadratic time MMD for the current data. Note that the type (biased/unbiased/incomplete) can be specified with set_statistic_type() method.
实现了 CKernelTwoSampleTest.
在文件 QuadraticTimeMMD.cpp 第 546 行定义.
Same as compute_statistic(), but with the possibility to perform on multiple kernels at once
| multiple_kernels | if true, and underlying kernel is K_COMBINED, method will be executed on all subkernels on the same data |
实现了 CKernelTwoSampleTest.
在文件 QuadraticTimeMMD.cpp 第 663 行定义.
computes a threshold based on current method for approximating the null-distribution. The threshold is the value that a statistic has to have in ordner to reject the null-hypothesis.
Not all methods for computing the p-value are compatible with all methods of computing the statistic (biased/unbiased/incomplete).
| alpha | test level to reject null-hypothesis |
重载 CTwoSampleTest .
在文件 QuadraticTimeMMD.cpp 第 801 行定义.
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protected |
Wrapper method for computing unbiased estimate of MMD^2
| m | number of samples from p |
| n | number of samples from q |
在文件 QuadraticTimeMMD.cpp 第 531 行定义.
Helper method to compute unbiased estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis
| m | number of samples from p |
| n | number of samples from q |
在文件 QuadraticTimeMMD.cpp 第 92 行定义.
Wrapper for computing variance estimate of the asymptotic distribution of the statistic (unbisaed/biased/incomplete) under null and alternative hypothesis (see class description for details)
在文件 QuadraticTimeMMD.cpp 第 598 行定义.
Same as compute_variance(), but with the possibility to perform on multiple kernels at once
| multiple_kernels | if true, and underlying kernel is K_COMBINED, method will be executed on all subkernels on the same data |
在文件 QuadraticTimeMMD.cpp 第 704 行定义.
| float64_t compute_variance_under_alternative | ( | ) |
Wrapper method for compute_variance()
在文件 QuadraticTimeMMD.cpp 第 658 行定义.
| float64_t compute_variance_under_null | ( | ) |
Wrapper method for compute_variance()
在文件 QuadraticTimeMMD.cpp 第 653 行定义.
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virtualinherited |
A deep copy. All the instance variables will also be copied.
在文件 SGObject.cpp 第 198 行定义.
Recursively compares the current SGObject to another one. Compares all registered numerical parameters, recursion upon complex (SGObject) parameters. Does not compare pointers!
May be overwritten but please do with care! Should not be necessary in most cases.
| other | object to compare with |
| accuracy | accuracy to use for comparison (optional) |
| tolerant | allows linient check on float equality (within accuracy) |
在文件 SGObject.cpp 第 618 行定义.
Approximates the null-distribution by the two parameter gamma distribution. It works in O(m^2) where m is the number of samples from each distribution. Its very fast, but may be inaccurate. However, there are cases where it performs very well. Returns parameters of gamma distribution that is fitted.
Called by compute_p_value() if null approximation method is set to MMD2_GAMMA.
Note that when being used for constructing a test, the provided statistic HAS to be the biased version (see paper for details). To use, set BIASED_DEPRECATED as statistic type. Note that m*Null-distribution is fitted, which is fine since the statistic is also m*MMD.
See Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011). A fast, consistent kernel two-sample test.
在文件 QuadraticTimeMMD.cpp 第 1032 行定义.
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virtualinherited |
在文件 KernelTwoSampleTest.h 第 86 行定义.
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inherited |
在文件 TwoSampleTest.h 第 127 行定义.
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inherited |
在文件 SGObject.cpp 第 498 行定义.
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Returns description of a given parameter string, if it exists. SG_ERROR otherwise
| param_name | name of the parameter |
在文件 SGObject.cpp 第 522 行定义.
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Returns index of model selection parameter with provided index
| param_name | name of model selection parameter |
在文件 SGObject.cpp 第 535 行定义.
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virtualinherited |
Getter for joint features, SG_REF'ed
被 CStreamingMMD 重载.
在文件 TwoSampleTest.cpp 第 171 行定义.
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returns the statistic type of this test statistic
实现了 CHypothesisTest.
在文件 QuadraticTimeMMD.h 第 286 行定义.
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virtualinherited |
If the SGSerializable is a class template then TRUE will be returned and GENERIC is set to the type of the generic.
| generic | set to the type of the generic if returning TRUE |
在文件 SGObject.cpp 第 296 行定义.
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virtualinherited |
Load this object from file. If it will fail (returning FALSE) then this object will contain inconsistent data and should not be used!
| file | where to load from |
| prefix | prefix for members |
在文件 SGObject.cpp 第 369 行定义.
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Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_POST is called.
| ShogunException | will be thrown if an error occurs. |
被 CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel , 以及 CExponentialKernel 重载.
在文件 SGObject.cpp 第 426 行定义.
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Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_PRE is called.
| ShogunException | will be thrown if an error occurs. |
被 CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool > , 以及 CDynamicObjectArray 重载.
在文件 SGObject.cpp 第 421 行定义.
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virtualinherited |
在文件 SGObject.cpp 第 262 行定义.
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virtualinherited |
Performs the complete two-sample test on current data and returns a p-value.
This is a wrapper that calls compute_statistic first and then calls compute_p_value using the obtained statistic. In some statistic classes, it might be possible to compute statistic and p-value in one single run which is more efficient. Therefore, this method might be overwritten in subclasses.
The method for computing the p-value can be set via set_null_approximation_method().
被 CStreamingMMD 重载.
在文件 HypothesisTest.cpp 第 113 行定义.
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Performs the complete two-sample test on current data and returns a binary answer wheter null hypothesis is rejected or not.
This is just a wrapper for the above perform_test() method that returns a p-value. If this p-value lies below the test level alpha, the null hypothesis is rejected.
Should not be overwritten in subclasses. (Therefore not virtual)
| alpha | test level alpha. |
在文件 HypothesisTest.cpp 第 121 行定义.
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prints all parameter registered for model selection and their type
在文件 SGObject.cpp 第 474 行定义.
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virtualinherited |
merges both sets of samples and computes the test statistic m_num_null_samples times. This version checks if a precomputed custom kernel is used, and, if so, just permutes it instead of re- computing it in every iteration.
重载 CTwoSampleTest .
被 CStreamingMMD 重载.
在文件 KernelTwoSampleTest.cpp 第 55 行定义.
Returns a set of samples of an estimate of the null distribution using the Eigen-spectrum of the centered kernel matrix of the merged samples of p and q. May be used to compute p-value (easy).
The estimate is computed as
\[ \frac{n_xn_y}{n_x+n_y}\hat{\eta}_{k,U}\rightarrow\sum_{l=1}^\infty \lambda_l\left(Z^2_l-1 \right) \]
where \({Z_l}\stackrel{i.i.d.}{\sim}\mathcal{N}(0,1)\) and \(\lambda_l\) are the eigenvalues of centered kernel matrix HKH.
kernel matrix needs to be stored in memory
Note that m*n/(m+n)*Null-distribution is returned, which is fine since the statistic is also m*n/(m+n)*MMD^2
Works well if the kernel matrix is NOT diagonal dominant. See Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011). A fast, consistent kernel two-sample test.
| num_samples | number of samples to draw |
| num_eigenvalues | number of eigenvalues to use to draw samples Maximum number of m+n-1 where m and n are the sizes of samples from p and q respectively. |
在文件 QuadraticTimeMMD.cpp 第 854 行定义.
| SGVector< float64_t > sample_null_spectrum_DEPRECATED | ( | index_t | num_samples, |
| index_t | num_eigenvalues | ||
| ) |
Returns a set of samples of an estimate of the null distribution using the Eigen-spectrum of the centered kernel matrix of the merged samples of p and q. May be used to compute p-value (easy).
The unbiased version uses
\[ t\text{MMD}_u^2[\mathcal{F},X,Y]\rightarrow\sum_{l=1}^\infty \lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}} -b_l\rho_y^{-\frac{1}{{2}}})^2-(\rho_x\rho_y)^{-1} \right) \]
where \(t=m+n\), \(\lim_{m,n\rightarrow\infty}m/t\rightarrow \rho_x\) and \(\rho_y\) likewise (equation 10 from [1]) and \(\lambda_l\) are estimated as \(\frac{\nu_l}{(m+n)}\), where \(\nu_l\) are the eigenvalues of centered kernel matrix HKH.
The biased version uses
\[ t\text{MMD}_b^2[\mathcal{F},X,Y]\rightarrow\sum_{l=1}^\infty \lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}}- b_l\rho_y^{-\frac{1}{{2}}})^2\right) \]
kernel matrix needs to be stored in memory
Note that (m+n)*Null-distribution is returned, which is fine since the statistic is also (m+n)*MMD: except when m and n are equal, then m*MMD^2 is returned
Works well if the kernel matrix is NOT diagonal dominant. See Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011). A fast, consistent kernel two-sample test.
| num_samples | number of samples to draw |
| num_eigenvalues | number of eigenvalues to use to draw samples Maximum number of m+n-1 where m and n are the sizes of samples from p and q respectively. It is usually safe to use a smaller number since they decay very fast, however, a conservative approach would be to use all (-1 does this). See paper for details. |
在文件 QuadraticTimeMMD.cpp 第 933 行定义.
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Save this object to file.
| file | where to save the object; will be closed during returning if PREFIX is an empty string. |
| prefix | prefix for members |
在文件 SGObject.cpp 第 314 行定义.
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Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_POST is called.
| ShogunException | will be thrown if an error occurs. |
被 CKernel 重载.
在文件 SGObject.cpp 第 436 行定义.
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Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_PRE is called.
| ShogunException | will be thrown if an error occurs. |
被 CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool > , 以及 CDynamicObjectArray 重载.
在文件 SGObject.cpp 第 431 行定义.
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在文件 SGObject.cpp 第 41 行定义.
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在文件 SGObject.cpp 第 46 行定义.
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在文件 SGObject.cpp 第 51 行定义.
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在文件 SGObject.cpp 第 56 行定义.
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在文件 SGObject.cpp 第 61 行定义.
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在文件 SGObject.cpp 第 66 行定义.
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在文件 SGObject.cpp 第 71 行定义.
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在文件 SGObject.cpp 第 76 行定义.
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在文件 SGObject.cpp 第 81 行定义.
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在文件 SGObject.cpp 第 86 行定义.
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在文件 SGObject.cpp 第 91 行定义.
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在文件 SGObject.cpp 第 96 行定义.
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在文件 SGObject.cpp 第 101 行定义.
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在文件 SGObject.cpp 第 106 行定义.
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在文件 SGObject.cpp 第 111 行定义.
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set generic type to T
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| m | number of samples from first distribution p |
在文件 TwoSampleTest.cpp 第 162 行定义.
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sets the method how to approximate the null-distribution
| null_approximation_method | method to use |
在文件 HypothesisTest.cpp 第 61 行定义.
| void set_num_eigenvalues_spectrum | ( | index_t | num_eigenvalues_spectrum | ) |
setter for number of eigenvalues to use in spectrum based p-value computation. Maximum is m_m+m_n-1
| num_eigenvalues_spectrum | number of eigenvalues to use to approximate null-distributrion |
在文件 QuadraticTimeMMD.cpp 第 1124 行定义.
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sets the number of permutation iterations for sample_null()
| num_null_samples | how often permutation shall be done |
在文件 HypothesisTest.cpp 第 67 行定义.
| void set_num_samples_spectrum | ( | index_t | num_samples_spectrum | ) |
setter for number of samples to use in spectrum based p-value computation.
| num_samples_spectrum | number of samples to draw from approximate null-distributrion |
在文件 QuadraticTimeMMD.cpp 第 1118 行定义.
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Setter for joint features
| p_and_q | joint features from p and q to set |
被 CStreamingMMD 重载.
在文件 TwoSampleTest.cpp 第 154 行定义.
| void set_statistic_type | ( | EQuadraticMMDType | statistic_type | ) |
| statistic_type | statistic type (biased/unbiased/incomplete) to use |
在文件 QuadraticTimeMMD.cpp 第 1130 行定义.
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A shallow copy. All the SGObject instance variables will be simply assigned and SG_REF-ed.
被 CGaussianKernel 重载.
在文件 SGObject.cpp 第 192 行定义.
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unset generic type
this has to be called in classes specializing a template class
在文件 SGObject.cpp 第 303 行定义.
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Updates the hash of current parameter combination
在文件 SGObject.cpp 第 248 行定义.
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io
在文件 SGObject.h 第 369 行定义.
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parameters wrt which we can compute gradients
在文件 SGObject.h 第 384 行定义.
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Hash of parameter values
在文件 SGObject.h 第 387 行定义.
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underlying kernel
在文件 KernelTwoSampleTest.h 第 121 行定义.
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defines the first index of samples of q
在文件 TwoSampleTest.h 第 139 行定义.
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model selection parameters
在文件 SGObject.h 第 381 行定义.
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Defines how the the null distribution is approximated
在文件 HypothesisTest.h 第 177 行定义.
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number of Eigenvalues for spectrum null-dstribution-approximation
在文件 QuadraticTimeMMD.h 第 479 行定义.
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number of iterations for sampling from null-distributions
在文件 HypothesisTest.h 第 174 行定义.
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number of samples for spectrum null-dstribution-approximation
在文件 QuadraticTimeMMD.h 第 476 行定义.
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protectedinherited |
concatenated samples of the two distributions (two blocks)
在文件 TwoSampleTest.h 第 136 行定义.
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parameters
在文件 SGObject.h 第 378 行定义.
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type of statistic (biased/unbiased/incomplete as well as deprecated versions of biased/unbiased)
在文件 QuadraticTimeMMD.h 第 484 行定义.
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parallel
在文件 SGObject.h 第 372 行定义.
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version
在文件 SGObject.h 第 375 行定义.
